Cubic critical, medium range

Average Error: 43.8 → 0.6

Time: 19.3s

Precision: binary64

Cost: 28352

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(\left(b + \sqrt{t_0}\right) \cdot \left(a \cdot -3\right)\right) \cdot \left(b \cdot b + t_0\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (/
    (fma 6.0 (* c (* a (* b b))) (* -9.0 (* (* c c) (* a a))))
    (* (* (+ b (sqrt t_0)) (* a -3.0)) (+ (* b b) t_0)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	return fma(6.0, (c * (a * (b * b))), (-9.0 * ((c * c) * (a * a)))) / (((b + sqrt(t_0)) * (a * -3.0)) * ((b * b) + t_0));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	return Float64(fma(6.0, Float64(c * Float64(a * Float64(b * b))), Float64(-9.0 * Float64(Float64(c * c) * Float64(a * a)))) / Float64(Float64(Float64(b + sqrt(t_0)) * Float64(a * -3.0)) * Float64(Float64(b * b) + t_0)))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(6.0 * N[(c * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-9.0 * N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 43.8

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Simplified 43.8

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}} \]
   Proof
   (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (fma.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 c (neg.f64 3))) (*.f64 b b))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 c (neg.f64 3)) a)) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c (*.f64 (neg.f64 3) a))) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (+.f64 (*.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 3 a)))) (*.f64 b b)))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 3 a)) c)) (*.f64 b b)))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 3 a)) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (+.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 b 1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (+.f64 (/.f64 b (Rewrite<= metadata-eval (/.f64 -1 -1))) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 b -1) -1)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 b)) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 -1/3 a)): 0 points increase in error, 31 points decrease in error
   (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) a)): 31 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 a)))): 0 points increase in error, 31 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 (/.f64 (neg.f64 b) -1) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (neg.f64 b) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 31 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 b)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
   (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 b -1)) -1)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
   (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> associate-/l*_binary64 (/.f64 b (/.f64 -1 -1)))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (/.f64 b (Rewrite=> metadata-eval 1))) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
   (+.f64 (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite=> /-rgt-identity_binary64 b)) (*.f64 (/.f64 -1 (*.f64 3 a)) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 0 points increase in error, 31 points decrease in error
   (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 -1 (*.f64 3 a)) (+.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))))): 0 points increase in error, 31 points decrease in error
   (*.f64 (/.f64 -1 (*.f64 3 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))): 31 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a))): 31 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
   (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error
   (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 31 points decrease in error

3. Applied egg-rr 42.8

   \[\leadsto \color{blue}{\frac{{b}^{4} - {\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}^{2}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}} \]

4. Taylor expanded in b around 0 0.7

   \[\leadsto \frac{\color{blue}{6 \cdot \left(c \cdot \left(a \cdot {b}^{2}\right)\right) + -9 \cdot \left({c}^{2} \cdot {a}^{2}\right)}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]

5. Simplified 0.6

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   Proof
   (/.f64 (fma.f64 6 (*.f64 c (*.f64 a (*.f64 b b))) (*.f64 -9 (*.f64 (*.f64 c c) (*.f64 a a)))) (*.f64 (*.f64 (*.f64 a -3) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) (+.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 6 (*.f64 c (*.f64 a (Rewrite<= unpow2_binary64 (pow.f64 b 2)))) (*.f64 -9 (*.f64 (*.f64 c c) (*.f64 a a)))) (*.f64 (*.f64 (*.f64 a -3) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) (+.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))))): 5 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 6 (*.f64 c (*.f64 a (pow.f64 b 2))) (*.f64 -9 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 a a)))) (*.f64 (*.f64 (*.f64 a -3) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) (+.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fma.f64 6 (*.f64 c (*.f64 a (pow.f64 b 2))) (*.f64 -9 (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))))) (*.f64 (*.f64 (*.f64 a -3) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) (+.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 6 (*.f64 c (*.f64 a (pow.f64 b 2)))) (*.f64 -9 (*.f64 (pow.f64 c 2) (pow.f64 a 2))))) (*.f64 (*.f64 (*.f64 a -3) (+.f64 b (sqrt.f64 (fma.f64 a (*.f64 c -3) (*.f64 b b))))) (+.f64 (*.f64 b b) (fma.f64 a (*.f64 c -3) (*.f64 b b))))): 5 points increase in error, 0 points decrease in error

6. Final simplification 0.6

   \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(a \cdot -3\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]

Alternatives

Alternative 1
Error	0.7
Cost	28288

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \frac{6 \cdot \left(c \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + -9 \cdot {\left(c \cdot a\right)}^{2}}{\left(\left(b + \sqrt{t_0}\right) \cdot \left(a \cdot -3\right)\right) \cdot \left(b \cdot b + t_0\right)} \end{array} \]

Alternative 2
Error	6.2
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b \cdot b - t_0}{a \cdot \left(b + \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	6.2
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b \cdot b - t_0\right)}{a \cdot \left(b + \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{a} \cdot \left(b \cdot b - t_0\right)}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	3.0
Cost	22272

\[\frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\mathsf{fma}\left(-12, a \cdot {b}^{3}, -3 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(\frac{a \cdot a}{b} \cdot 2.25\right)\right) + c \cdot \left(a \cdot \left(-9 \cdot \left(a \cdot b\right)\right)\right)\right)\right)} \]

Alternative 6
Error	6.4
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{-0.3333333333333333}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	6.3
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{\frac{a}{-0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	6.6
Cost	14980

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + 1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 9
Error	6.3
Cost	14980

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot 1.125\right)\right) + -0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	6.6
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -0.13:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(a \cdot 1.125\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\right) + -0.3333333333333333 \cdot \left(c \cdot \frac{1.5}{b}\right)\\ \end{array} \]

Alternative 11
Error	6.4
Cost	1472

\[-0.3333333333333333 \cdot \left(\left(a \cdot 1.125\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\right) + -0.3333333333333333 \cdot \left(c \cdot \frac{1.5}{b}\right) \]

Alternative 12
Error	12.1
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2022349 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))